Question

State whether the given sentence is true or false. If false, replace the underlined term to make a true sentence.

The graph of a quadratic function has a maximum if the coefficient of the $x^2$is positive.

Short answer

The sentence, “The graph of a quadratic function has a maximum if the coefficient of the $x^2$ is positive” is false.

The correct replacement for the underlined term to make a true sentence is minimum.

Step-by-step solution

Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y=a{x}^2+bx+c$, where, $a\ne 0$is called the standard form of the quadratic function.

Step 2. Define the maximum point of a function.

The maximum value of a function is the place where the function reaches its highest point.

Step 3. Check whether the given sentence is true or false.

The graph of the parabola $y=a{x}^2+bx+c$,

Opens upward and has a minimum value at $x=-\frac{b}{2a}$, when $a>0$.

Opens downward and has a maximum value at $x=-\frac{b}{2a}$, when $a<0$.

Therefore, the sentence, “The graph of a quadratic function has a maximum if the coefficient of the$x^2$ is positive” is false.

The correct replacement for the underlined term to make a true sentence is minimum.